ManyPrimates pilot mixed modeling

library(tidyverse)
library(knitr)
library(sjPlot)
library(gridExtra)
library(lme4)
library(emmeans)
library(car) # for vif
library(bbmle) # for AICtab
library(broom) # for glance
theme_set(ggthemes::theme_few())

Summary

Mixed modeling with all relevant variables predicting accuracy

From the preregistration, the mixed model was specified thusly:

correct ~ delay * age + 
          task_experience + cup_distance + board_size + trial +
          (1 + delay + trial | site/subject/block/hiding_location ) + 
          (1 + task_experience + cup_distance + board_size + trial + delay | species)

In the dataframe, subject_site = subject, and norm_age should be used for age.

Model as pre-registered has too many random effects

Error: number of observations (=6246) < number of random effects (=10608) for term (1 + delay + trial | hiding_location:(block:(subject_site:site))); the random-effects parameters are probably unidentifiable

Pruning random effects in the following order (from preregistration):

• Remove correlations between random effects
• Remove random slopes (in the following order)
  • species
  • hiding_location
  • block
  • subject

Model only converges once we take out hiding_location. After doing so, the other random effects (correlation, site, species) can be put back in.

The model below converges. Model output is saved in 06_mp_model_v2.rds

correct ~ delay * norm_age + 
          task_experience + cup_distance + board_size + trial + 
          (1 + delay + trial | site/subject_site) + 
          (1 + task_experience + cup_distance + board_size + trial + delay | species)

Reduced model

After pruning random effects with little variability and removing board_size, which covaried with cup_distance, the reduced model has the following structure. It is saved in 06_mp_3_model3_v2.rds

correct ~ delay * norm_age + 
          task_experience + cup_distance + board_size + trial + 
          (1 + delay | site/subject_site) + 
          (1 + delay | species)

---

Data prep

Data import

mp_data <- read.csv("../data/merged_data/01_manyprimates_pilot_merged_data_v2.csv")

Prepare code for pre-registered mixed modeling

• center cup_distance, board_size and trial
• filter out spider monkey. Only one data point so far, therefore this is not worth including to explode the number of random effects

model.data <- mp_data %>%
  filter(species != "black_faced_spider_monkey") %>%
  mutate_at(vars(cup_distance, board_size, trial), funs(scale(.)[, 1])) %>%
  mutate(hiding_location = factor(hiding_location),
         delay = fct_relevel(delay, "short"))

Model 1

The model takes a while to run. Run next line to load model output from previous run with structure below.

mm.1 <- readRDS("06_mp_model1.rds")

mm.1 <- glmer(correct ~ delay * norm_age +
               task_experience + cup_distance + board_size + trial +
               (1 + delay + trial | site/subject_site/block) +
               (1 + task_experience + cup_distance + board_size + trial + delay | species)
             , data = model.data
             , family = binomial
             , control = glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e5))
             )

saveRDS(mm.1, "06_mp_model1.rds")

Some diagnostics

• examining Cholesky decomposition

theta <- getME(mm.1, "theta")
diag.element <- getME(mm.1, "lower") == 0
any(theta[diag.element] < 1e-5)

[1] TRUE

Model summary

Confirm model structure

# mm.1@call
formula(mm.1)

correct ~ delay * norm_age + task_experience + cup_distance + 
    board_size + trial + (1 + delay + trial | site/subject_site) + 
    (1 + task_experience + cup_distance + board_size + trial + 
        delay | species)

glance(mm.1) %>% kable(digits = 2)

sigma	logLik	AIC	BIC	deviance	df.residual
1	-3388.28	6892.57	7283.47	6365.11	6188

Random effects

fmt <- function(num, digits) return(round(num, digits))
VarCorr(mm.1) %>% print(formatter = fmt, digits = 3) # comp = c("Variance", "Std.Dev.")

 Groups            Name               Std.Dev. Corr                               
 subject_site:site (Intercept)        0.862                                       
                   delaylong          0.646    -0.91                              
                   delaymedium        0.528    -0.85  0.99                        
                   trial              0.093    -0.31  0.61  0.71                  
 site              (Intercept)        0.917                                       
                   delaylong          0.512    -1.00                              
                   delaymedium        0.627    -1.00  1.00                        
                   trial              0.084     1.00 -1.00 -1.00                  
 species           (Intercept)        0.835                                       
                   task_experienceyes 0.173    -1.00                              
                   cup_distance       0.012     1.00 -1.00                        
                   board_size         0.232    -1.00  1.00 -1.00                  
                   trial              0.021    -1.00  1.00 -1.00  1.00            
                   delaylong          0.494    -1.00  1.00 -1.00  1.00  1.00      
                   delaymedium        0.436    -1.00  1.00 -1.00  1.00  1.00  1.00

Fixed effects

CIs

mm.1.ci <- confint(mm.1, method = "Wald") %>% # bootstrap these later
  as.data.frame %>%
  rownames_to_column %>%
  filter(complete.cases(.)) %>%
  rename(LL = `2.5 %`, UL = `97.5 %`) %>%
  mutate(OR_LL = exp(LL), OR_UL = exp(UL))

coef(summary(mm.1)) %>%
  as.data.frame %>%
  rownames_to_column() %>%
  mutate(OR = exp(Estimate)) %>%
  left_join(mm.1.ci, by = "rowname") %>%
  select(rowname, OR, OR_LL, OR_UL, Estimate, LL, UL, everything()) %>%
  kable(digits = 3)

rowname	OR	OR_LL	OR_UL	Estimate	LL	UL	Std. Error	z value	Pr(>|z|)
(Intercept)	4.135	1.703	10.040	1.420	0.532	2.307	0.453	3.136	0.002
delaylong	0.284	0.169	0.477	-1.257	-1.775	-0.740	0.264	-4.763	0.000
delaymedium	0.369	0.213	0.638	-0.997	-1.545	-0.450	0.279	-3.570	0.000
norm_age	0.987	0.796	1.224	-0.013	-0.228	0.202	0.110	-0.117	0.907
task_experienceyes	1.234	0.784	1.940	0.210	-0.243	0.663	0.231	0.908	0.364
cup_distance	1.554	1.253	1.927	0.441	0.226	0.656	0.110	4.014	0.000
board_size	1.779	1.263	2.506	0.576	0.233	0.919	0.175	3.296	0.001
trial	1.046	0.953	1.148	0.045	-0.048	0.138	0.047	0.948	0.343
delaylong:norm_age	1.049	0.854	1.289	0.048	-0.158	0.253	0.105	0.454	0.650
delaymedium:norm_age	1.082	0.887	1.320	0.079	-0.120	0.278	0.102	0.775	0.438

corr <- cov2cor(vcov(mm.1)) %>% as.matrix %>% round(2)
corr[upper.tri(corr, diag = T)] <- ""
colnames(corr) <- 1:10
rownames(corr) <- str_c(1:10, " ", rownames(corr))

corr %>% as.data.frame %>% select(-10) %>% rownames_to_column

Pairwise contrasts for delay

based on estimated marginal means

Note. This wasn’t in the preregistration.

emmeans(mm.1, pairwise ~ delay, type = "response")$contrasts

 contrast       odds.ratio         SE  df z.ratio p.value
 short / long    3.5160746 0.92828519 Inf   4.762  <.0001
 short / medium  2.7103806 0.75716209 Inf   3.569  0.0010
 long / medium   0.7708541 0.07335892 Inf  -2.735  0.0172

Results are averaged over the levels of: task_experience 
P value adjustment: tukey method for comparing a family of 3 estimates 
Tests are performed on the log odds ratio scale 

Model 1 plots

Fixed effects

plot_model(mm.1, title = "Fixed Effects", order.terms = c(7, 4, 3:1, 9:8, 5, 6),
           width = .3, show.values = T, value.size = 2.5, value.offset = .3) +
  geom_hline(yintercept = 1, lty = 2) +
  ylim(0, 3)

Random effects

ranef.plots <- plot_model(mm.1, type = "re", sort.est = "(Intercept)")

Subject/Site

ranef.plots[[1]]

Site

ranef.plots[[2]]

Species

ranef.plots[[3]]

---

Pruning the model

• remove trial random slopes within species as the estimates in the previous models were essentially 0
• remove trial from the random slopes for subject/site for the same reason

correct ~ delay * norm_age + 
          task_experience + cup_distance + board_size + trial +
          (1 + delay | site/subject_site ) +         
          (1 + task_experience + cup_distance + board_size + delay | species)

Check colinearity in the previous model

col.mm1 <- glm(correct ~ delay + norm_age +
                 task_experience + cup_distance + board_size + trial
               , data = model.data
               , family = binomial)
vif(col.mm1)

                    GVIF Df GVIF^(1/(2*Df))
delay           1.007876  2        1.001963
norm_age        1.072757  1        1.035740
task_experience 1.056383  1        1.027805
cup_distance    1.323990  1        1.150648
board_size      1.296698  1        1.138726
trial           1.001067  1        1.000533

No signs of high colinearity.

Check levels of random effects

Check how many different levels there are within each random effect

source("diagnostic_fcns.r")
overview <- fe.re.tab("correct ~ delay + task_experience + board_size + cup_distance + trial", "species", data = model.data)
overview$summary

$`delay_within_species (factor)`

 3 
11 

$`task_experience_within_species (factor)`

1 2 
9 2 

$`board_size_within_species (covariate)`

1 2 4 
5 5 1 

$`cup_distance_within_species (covariate)`

1 2 4 
6 4 1 

$`trial_within_species (covariate)`

36 
11 

This suggests that, within species, random slopes for task_experience does not make much sense as most species have only 1 level. Same is true for cup_distance and board_size. Indeed, the model summary and random effects plot for species confirm that there is little variability in these estimates (they’re close to zero). Therefore they are removed.

correct ~ delay * norm_age + 
          task_experience + cup_distance + board_size + trial +
          (1 + delay + trial | site/subject_site ) +         
          (1 + delay | species)

Model 2

mm.2 <- readRDS("06_mp_model2.rds")

mm.2 <- glmer(correct ~ delay * norm_age +
              task_experience + cup_distance + board_size  + trial +
              (1 + delay | site/subject_site) +
              (1 + delay | species)
              , data = model.data
              , family = binomial
              , control = glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e5))
              )

saveRDS(mm.2, "06_mp_model2.rds")

Model summary

Confirm model structure

formula(mm.2)

correct ~ delay * norm_age + task_experience + cup_distance + 
    board_size + trial + (1 + delay | site/subject_site) + (1 + 
    delay | species)

glance(mm.2) %>% kable(digits = 2)

sigma	logLik	AIC	BIC	deviance	df.residual
1	-3391.26	6838.52	7027.23	6389.41	6218

LRT

drop1(mm.2, test = 'Chisq')

Single term deletions

Model:
correct ~ delay * norm_age + task_experience + cup_distance + 
    board_size + trial + (1 + delay | site/subject_site) + (1 + 
    delay | species)
                Df    AIC     LRT   Pr(Chi)    
<none>             6838.5                      
task_experience  1 6836.5  0.0074 0.9314327    
cup_distance     1 6848.6 12.0934 0.0005060 ***
board_size       1 6847.9 11.3967 0.0007357 ***
trial            1 6837.1  0.5912 0.4419742    
delay:norm_age   2 6835.3  0.8225 0.6628194    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Random effects

VarCorr(mm.2) %>% print(comp = c("Variance", "Std.Dev."), formatter = fmt, digits = 3)

 Groups            Name        Variance Std.Dev. Corr       
 subject_site:site (Intercept) 0.734    0.857               
                   delaylong   0.416    0.645    -0.90      
                   delaymedium 0.261    0.511    -0.85  1.00
 site              (Intercept) 1.087    1.043               
                   delaylong   0.367    0.606    -1.00      
                   delaymedium 0.456    0.675    -1.00  1.00
 species           (Intercept) 0.481    0.694               
                   delaylong   0.214    0.462    -1.00      
                   delaymedium 0.157    0.396    -1.00  1.00

Fixed effects

CIs

mm.2.ci <- readRDS("06_mp_model2_ci.rds")

# Bootstrap function by Roger Mundry @ MPI EVA
source("boot_glmm.r")

mm.2.ci <- boot.glmm.pred(model.res = mm.2, excl.warnings = F, nboots = 1000,
                         para = F, resol = 100, level = 0.95, use = NULL,
                         circ.var.name = NULL, circ.var = NULL, use.u = F,
                         n.cores = c("all-1", "all"), save.path = NULL)

saveRDS(mm.2.ci, "06_mp_model2_ci.rds")

coef(summary(mm.2)) %>%
  as.data.frame %>%
  rownames_to_column() %>%
  mutate(OR = exp(Estimate)) %>%
  cbind(mm.2.ci$ci.estimates[, 2:3]) %>%
  select(rowname, OR, Estimate, X2.5., X97.5., everything()) %>%
  kable(digits = 3)

	rowname	OR	Estimate	X2.5.	X97.5.	Std. Error	z value	Pr(>|z|)
(Intercept)	(Intercept)	5.805	1.759	0.978	2.541	0.448	3.925	0.000
delaylong	delaylong	0.240	-1.426	-1.943	-0.939	0.280	-5.092	0.000
delaymedium	delaymedium	0.313	-1.161	-1.663	-0.665	0.281	-4.130	0.000
norm_age	norm_age	0.992	-0.008	-0.212	0.211	0.109	-0.078	0.938
task_experienceyes	task_experienceyes	1.016	0.015	-0.282	0.350	0.173	0.089	0.929
cup_distance	cup_distance	1.462	0.380	0.169	0.613	0.103	3.677	0.000
board_size	board_size	1.472	0.386	0.198	0.593	0.094	4.097	0.000
trial	trial	1.024	0.023	-0.037	0.086	0.030	0.774	0.439
delaylong:norm_age	delaylong:norm_age	1.052	0.051	-0.162	0.249	0.106	0.479	0.632
delaymedium:norm_age	delaymedium:norm_age	1.093	0.089	-0.117	0.286	0.101	0.883	0.377

Pairwise contrasts for delay

based on estimated marginal means

emmeans(mm.2, pairwise ~ delay)$contrasts

 contrast         estimate        SE  df z.ratio p.value
 short - long    1.4257372 0.2800475 Inf   5.091  <.0001
 short - medium  1.1604769 0.2810297 Inf   4.129  0.0001
 long - medium  -0.2652602 0.0865917 Inf  -3.063  0.0062

Results are averaged over the levels of: task_experience 
Results are given on the log odds ratio (not the response) scale. 
P value adjustment: tukey method for comparing a family of 3 estimates 

code to bootstrap these CIs (confint(emmeans(...) gets Wald CIs)

boot.pairs <- function(m) suppressMessages(summary(pairs(emmeans(m, "delay")))$estimate)
boot.emm <- function(m, n = 1000) {
  boo <- bootMer(m, FUN = boot.pairs, nsim = n, parallel = "multicore", ncpus = 4)
  ci <- sapply(1:3, function(j) quantile(boo$t[,j], prob = c(.025, .975), na.rm = T))
  ci <- t(rbind(boo$t0, ci))
  rownames(ci) <- c("short vs. long", "short vs. medium", "medium vs. long")
  return(ci)
}

This took ~13 hours to run on 4 cores on my machine (jw). Skip this chunk and run the next one to load CIs from previous run.

system.time(mm2.emm.ci <- boot.emm(mm.2, n = 1000))
saveRDS(mm2.emm.ci, "06_mp_model2_emm_ci.rds")

      user     system    elapsed 
130648.652   1127.768  47989.548

mm2.emm.ci <- readRDS("06_mp_model2_emm_ci.rds")
round(mm2.emm.ci, 2)

                        2.5% 97.5%
short vs. long    1.43  0.95  2.01
short vs. medium  1.16  0.67  1.73
medium vs. long  -0.27 -0.44 -0.10

odds ratios

round(exp(mm2.emm.ci), 2)

                      2.5% 97.5%
short vs. long   4.16 2.58  7.46
short vs. medium 3.19 1.94  5.65
medium vs. long  0.77 0.64  0.90

Model 2 plots

Fixed effects

fe.2 <- plot_model(mm.2, title = "A. Fixed Effects", 
                   axis.labels = c("Trial", "Task Experience (yes)","Board Size (cm)", 
                                   "Cup Distance (cm)", "Normed Age x Delay\n(short vs. long)",
                                   "Normed Age x Delay\n(short vs. medium)", "Normed Age", 
                                   "Delay (short vs. long)", "Delay (short vs. medium)"), 
                   order.terms = c(2:1, 3, 9:8, 5, 6, 4, 7), wrap.labels = F,
                   width = .3, show.values = T, value.size = 2.5, value.offset = .3) +
  facet_wrap(~ "") +
  # geom_hline(yintercept = 1, lty = 2) +
  scale_y_continuous(trans = "log", limits = c(.1, 5.5), breaks = c(.1, .2, .5, 1, 2, 5))
fe.2 + theme(plot.margin = unit(c(.5, 7, .5, .5), "cm"))

ggsave("../graphs/05_forestplot.png", fe.2, width = 3, height = 2.5, scale = 2)

Random effects

ranef.plots2 <- plot_model(mm.2, type = "re", sort.est = "(Intercept)")

Subject/Site

ranef.plots2[[1]]

Site

ranef.plots2[[2]]

Species

ranef.plots2[[3]]

Model 3

• further remove subject/site random effects to look at species differences (from preregistration)

mm.3 <- glmer(correct ~ delay * norm_age +
                task_experience + cup_distance + trial +
                (1 + delay | species)
              , data = model.data
              , family = binomial
              , control = glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e5))
        )

Random effects

phylo <- read_csv("../data/species_data.csv") %>% select(species, species_formatted, phylo)

plot.data <- get_model_data(mm.3, type = "re", show.values = T) %>%
  left_join(phylo, by = c("term" = "species")) %>%
  rename(species = species_formatted) %>%
  mutate(
    facet = fct_rev(facet),
    facet = fct_recode(facet, "Intercept" = "species (Intercept)",
                       "Delay (short vs. long)" = "delaylong", 
                       "Delay (short vs. medium)" = "delaymedium")
    )

Column `term`/`species` joining factor and character vector, coercing into character vector

sorted <- filter(plot.data, facet == "Intercept") %>% arrange(estimate) %>% with(species)
plot.data <- mutate(plot.data, species = factor(species, levels = sorted))

re.3 <- ggplot(plot.data, aes(x = species, y = estimate, col = group)) +
  facet_grid(~ facet) +
  geom_hline(yintercept = 1, col = "grey90", size = 1.125) +
  # geom_hline(yintercept = 1, lty = 2) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high), width = .3) +
  geom_point(size = 2.5) +
  geom_text(aes(label = p.label), nudge_x = .3, size = 2.5, show.legend = F) +
  scale_y_continuous("Odds Ratios", trans = "log", breaks = c(.1, .2, .5, 1, 2, 5,10,20,40)) +
  scale_colour_brewer(palette = "Set1") +
  coord_flip(ylim = c(.1, 42.5)) + xlab("") +
  guides(col = "none") +
  ggtitle("B. Species Random Effects")

mat <- matrix(c(rep(1, 3), rep(2, 7)), nrow = 1)
grid.arrange(fe.2, re.3, layout_matrix = mat)

ggsave("../graphs/05_forestplot_fe_re.png", arrangeGrob(fe.2, re.3, layout_matrix = mat), width = 8, height = 3, scale = 1.8)
ggsave("../graphs/Fig3.tiff", arrangeGrob(fe.2, re.3, layout_matrix = mat), width = 8, height = 3, scale = 1.8, type = "cairo", compression = "lzw")

Model comparison

We’re looking for the lowest AIC(c) as the model with the ‘best fit’ with a reasonable number of parameters. (Too many are penalized by AIC as one way to address overfitting.)

Indeed, the reduced model seems to do a better job of striking that balance between fitting the data with fewer parameters.

AICctab(mm.1, mm.2, mm.3, logLik = T, weights = T)

     dLogLik dAICc df weight
mm.2 109.9     0.0 28 1     
mm.1 112.9    54.9 58 <0.001
mm.3   0.0   193.6 15 <0.001

anova(mm.1, mm.2, mm.3)

Data: model.data
Models:
mm.3: correct ~ delay * norm_age + task_experience + cup_distance + 
mm.3:     trial + (1 + delay | species)
mm.2: correct ~ delay * norm_age + task_experience + cup_distance + 
mm.2:     board_size + trial + (1 + delay | site/subject_site) + (1 + 
mm.2:     delay | species)
mm.1: correct ~ delay * norm_age + task_experience + cup_distance + 
mm.1:     board_size + trial + (1 + delay + trial | site/subject_site) + 
mm.1:     (1 + task_experience + cup_distance + board_size + trial + 
mm.1:         delay | species)
     Df    AIC    BIC  logLik deviance    Chisq Chi Df Pr(>Chisq)    
mm.3 15 7032.3 7133.4 -3501.1   7002.3                               
mm.2 28 6838.5 7027.2 -3391.3   6782.5 219.7490     13     <2e-16 ***
mm.1 58 6892.6 7283.5 -3388.3   6776.6   5.9521     30          1    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Difference in regression coefficients

Difference

coef1 <- coef(summary(mm.1))[c(2, 3, 6), 1]
coef2 <- coef(summary(mm.2))[c(2, 3, 6), 1]
coef2 - coef1

   delaylong  delaymedium cup_distance 
 -0.16840056  -0.16341754  -0.06131301 

Difference in odds ratios

exp(coef2) - exp(coef1)

   delaylong  delaymedium cup_distance 
 -0.04407273  -0.05561161  -0.09242183